Cayley, Sylvester, and Early Matrix Theory

Abstract

2008 marks the 150th anniversary of “A Memoir on the Theory of Matrices” by Arthur Cayley (1821–1895) [3]—the first paper on matrix algebra. Prior to this paper the theory of determinants was well developed, and Cauchy had shown that the eigenvalues of a real matrix are real (in the context of quadratic forms). Yet the idea that an array of numbers had algebraic properties that merited study in their own right was first put forward by Cayley. The term “matrix” had already been coined in 1850 by James Joseph Sylvester (1814– 1897) [20]. The names of Cayley and Sylvester are of course well known to any student of linear algebra and matrix analysis, through eponymous objects such as the Cayley–Hamilton theorem, the Cayley transformation, Sylvester’s inertia theorem, and the Sylvester equation. The lives of these two mathematicians have been well documented during the century or so since their deaths, for example in [1, Chap, 21], [13], [14], [15], [18]. So what is the significance of these two new biographies, both published in 2006 by Johns Hopkins University Press? There are two answers. First, both authors are the leading experts on the respective mathematicians, having spent much of their lives studying their work, their voluminous correspondence, and their place in the contemporary world of mathematics. Consequently, both biographies are authoritative and comprehensive, and they include new information that has come to light only in recent years. Second, as their titles indicate, both biographies put the lives of Cayley and Sylvester into their proper historical context, thereby giving insight into what it was like to be a mathematician in the 19th century and explaining the particular problems that Cayley, and more particularly Sylvester because of his Jewish religion, faced in carving out a career as a research mathematician.